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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

54 IV + ~ . Typical

54 IV + ~ . Typical phase portrait in P1 in the presence of type 4 equilibria (one pair named IV + , the conjugates not named). We indicate the fourth and fifth order terms which are required in the equations restricted to the subspace V 2 (1=2 modes) for the analysis of type 4 equilibria: Ck(Y,Y).[hlIYII 2 + h'p(Y)], where Ck(Y,Y) are the quadratic equivariants defined in (6) and p(Y) is the cubic invariant polynomial p(Y) = y03- 3Y0y.ly 1 - 6Y0y.2y 2 + 3q-6 (y12Y_2+Y_12y2). Now, thanks to the presence of 1 =1 modes, one can find values of ~-1, ~-2 and c at which the critical m=2 modes (for the 0(2) action) are superposed to critical m=l modes, resulting in a quadruple semi-simple 0 eigenvalue. Then the problem enters into the frame of the study of Armbruster et al. [1988], leading to the existence and stability of a heteroclinic cycle connecting the type 4 equilibria. More precisely, critical values of the parameters can be found at which the eigenvalues at the type 1 solutions vanish in the directions {xl,xq,y2,y_2} and the other eigenvalues not forced to 0 by the group action have negative real part. Then a center manifold reduction can be performed at this point in the 5-dimensional space Fix(Z~ (which contains the critical m=l and 2 modes, see rigA) allowing to derive the suitable conditions by identification with the conditions given in

55 Armbruster et al. [1988]. This work is done in Armbruster, Chossat [1989]. We show in figure 11 the simulation of such a heteroclinic cycle. 4.2. Asymptotic stability Table 2. Heteroclinic cycles which generically exist under hypothesis H1. Name Connexions o~-~ oc ~ ~ in P2 [~ , (o~',(x") in P1 cx-type 2 o~ ) type 2 in P2 type 2----* (o~',ot") in S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c~-limit cycle c~ , limit cycle in P2 limit cycle , (c~',c~") in S type 4-type 4 type 4--'* type 4 in Fix(Z2-) In the case of a heteroclinic cycle living on a two-dimensional invariant normally hyperbolic manifold, a necessary and sufficient condition for the asymptotic stability of the HC is that along this manifold, the product of the stable eigenvalues be larger in modulus than the product of the unstable ones (Dos Reis [1984]). The situation is less clear when more than two dimensions are involved in the dynamics associated to the HC. A sufficient condition for stability was derived in Melbourne et al. [1988] in the following way: suppose that the HC exists in the context of proposition 1. For each equilibrium Xj we consider the eigenvalues of the tinearized operator in the 3 dimensional space

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